Communications in Mathematical Physics

, Volume 279, Issue 1, pp 117–146

The Effect of Disorder on Polymer Depinning Transitions

Article

Abstract

We consider a polymer, with monomer locations modeled by the trajectory of a Markov chain, in the presence of a potential that interacts with the polymer when it visits a particular site 0. We assume that probability of an excursion of length n is given by \(n^{-c}\varphi(n)\) for some 1 < c < 2 and slowly varying \(\varphi\) . Disorder is introduced by having the interaction vary from one monomer to another, as a constant u plus i.i.d. mean-0 randomness. There is a critical value of u above which the polymer is pinned, placing a positive fraction (called the contact fraction) of its monomers at 0 with high probability. To see the effect of disorder on the depinning transition, we compare the contact fraction and free energy (as functions of u) to the corresponding annealed system. We show that for c > 3/2, at high temperature, the quenched and annealed curves differ significantly only in a very small neighborhood of the critical point—the size of this neighborhood scales as \(\beta^{1/(2c-3)}\), where β is the inverse temperature. For c < 3/2, given \(\epsilon > 0\), for sufficiently high temperature the quenched and annealed curves are within a factor of \(1-\epsilon\) for all u near the critical point; in particular the quenched and annealed critical points are equal. For c = 3/2 the regime depends on the slowly varying function \(\varphi\).

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References

  1. 1.
    Alexander K.S. and Sidoravicius V. (2006). Pinning of polymers and interfaces by random potentials. Ann. Appl. Probab. 16: 636–669 MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Alexander, K.S.: Ivy on the ceiling: first-order polymer depinning transitions with quenched disorder. http://arxiv.org/list/:math.PR/0612625, (2006)
  3. 3.
    Azuma K. (1967). Weighted sums of certain dependent random variables. Tohoku Math. J. 19: 357–367 MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bodineau T. and Giacomin G. (2004). On the localization transition of random copolymers near selective interfaces. J. Stat. Phys. 117: 801–818 MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Caravenna F., Giacomin G. and Gubinelli M. (2006). A numerical approach to copolymers at selective interfaces. J. Stat. Phys. 122: 799–832 MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Derrida B., Hakim V. and Vannimenus J. (1992). Effect of disorder on two-dimensional wetting. J. Stat. Phys. 66: 1189–1213 MATHCrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Doney R.A. (1997). One-sided local large deviation and renewal theorems in the case of infinite mean. Probab. Theory Rel. Fields 107: 451–465 MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Fisher M.E. (1984). Walks, walls, wetting and melting. J. Stat. Phys. 34: 667–729 MATHCrossRefADSGoogle Scholar
  9. 9.
    Forgacs G., Luck J.M., Nieuwenhuizen Th. M. and Orland H. (1988). Exact critical behavior of two-dimensional wetting problems with quenched disorder. J. Stat. Phys. 51: 29–56 MATHCrossRefADSGoogle Scholar
  10. 10.
    Galluccio S. and Graber R. (1996). Depinning transition of a directed polymer by a periodic potential: a d-dimensional solution. Phys. Rev. E 53: R5584–R5587 Google Scholar
  11. 11.
    Giacomin G. (2007). Random Polymer Models. Cambridge University Press, Cambridge MATHGoogle Scholar
  12. 12.
    Giacomin G. and Toninelli F.L. (2006). The localized phase of disordered copolymers with adsorption. Alea 1: 149–180 MATHMathSciNetGoogle Scholar
  13. 13.
    Giacomin G. and Toninelli F.L. (2006). Smoothing effect of quenched disorder on polymer depinning transitions. Commun. Math. Phys. 266: 1–16 MATHCrossRefADSMathSciNetGoogle Scholar
  14. 14.
    Gotcheva V. and Teitel S. (2001). Depinning transition of a two-dimensional vortex lattice in a commensurate periodic potential. Phys. Rev. Lett. 86: 2126–2129 CrossRefADSGoogle Scholar
  15. 15.
    Monthus C. (2000). On the localization of random heteropolymers at the interface between two selective solvents. Eur. Phys. J. B 13: 111–130 ADSGoogle Scholar
  16. 16.
    Morita T. (1964). Statistical mechanics of quenched solid solutions with applications to diluted alloys. J. Math. Phys. 5: 1401–1405 CrossRefADSGoogle Scholar
  17. 17.
    Mukherji S. and Bhattacharjee S.M. (1993). Directed polymers with random interaction: An exactly solvable case. Phys. Rev. E 48: 3483–3496 CrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Naidenov A. and Nechaev S. (2001). Adsorption of a random heteropolymer at a potential well revisited: location of transition point and design of sequences. J. Phys. A: Math. Gen. 34: 5625–5634 MATHCrossRefADSMathSciNetGoogle Scholar
  19. 19.
    Nechaev S. and Zhang Y.-C. (1995). Exact solution of the 2D wetting problem in a periodic potential. Phys. Rev. Lett. 74: 1815–1818 CrossRefADSGoogle Scholar
  20. 20.
    Nelson D.R. and Vinokur V.M. (1993). Boson localization and correlated pinning of superconducting vortex arrays. Phys. Rev. B 48: 13060–13097 CrossRefADSGoogle Scholar
  21. 21.
    Orlandini E., Rechnitzer A. and Whittington S.G. (2002). Random copolymers and the Morita aproximation: polymer adsorption and polymer localization. J. Phys. A: Math. Gen. 35: 7729–7751 MATHCrossRefADSMathSciNetGoogle Scholar
  22. 22.
    Seneta, E.: Regularly Varying Functions. Lecture Notes in Math. 508. Berlin: Springer-Verlag, 1976Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of Mathematics KAP108University of Southern CaliforniaLos AngelesUSA

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